how to read CI

Note from reading 'Inference by Eye'. The interpretation of CI figures does not only require the knowledge of what are plotted (SE vs SD vs CI) but alos require the knowledge of experiement design / analysis context (whether it shows group means of independent samples vs pre-post means of repeated measures vs meta analysis). It is important to understand what effect or comparison is the major interest.

CI is just one from an infinite sequence: if the experiment ware repeated many times and a CI calculated for each, in the long run 95% of the CI will include the true mean. Equivalentlyl, a research who routinely reports 95% CI can expect over a lifetime that about 95% of those intervals will catpure the true mean. To interpret CI: CI is a range of plausible values for mean; values outside the CI are relatively implausible.

The width of CI is the largest error of estimation we are likely to make.

for a comparison of two independent means, p<=0.05 when the overlap of the 95%CI is no more than about half the average width of CI, that is, when proportion overlap is about half. In addition, p<=.01 when the two CI do not overlap. If we see SE, and consider the relationship between SE and 95% CI, P<=0.05 when the gap between the SE bars is at lease about the size of the average SE(of the 2 groups). This rule does not work at all for paired data, because the width of CI for the difference is sensitive to the correlation between the pairs; positive correlation will reduce the width of CI for the mean difference.

optimality of Likelihood ratio

From wikipedia:

the Neyman–Pearson lemma describes the existence and uniqueness of the likelihood ratio as a uniformly most powerful test in certain contexts.

I recently case across its applications in 2 occasions

• showing logistic regression is the optimal test to combine several predictors for a binary outcome.
• showing bonferoni is optimal test for the global null against a sparse alternative (only 1 alternative hypothesis is true among many many true nulls), and fisher's combination test is optimal against the alternative of small distributed effects. In this example, it becomes obvious that optimality depends on the likelihood of alternative

The first example is suspicious because I know throwing all predictors in a linear model often does not improve model perforance obviously. By the way, the link between odds ratios from individual predictors and overall performance is weak (from a publication of Pepe in the follow year, though the paper was mostly about the disconnect in univariate models, covering the increment value of markers near the end: a marker with a statistically significant OR adds little to the AUC)

Initially, I was wondering whether this is because Neyman–Pearson lemma is developed for simple hypothesis. A little investigation suggests that there is no magic property of a simple hypothesis vs a composite one; the key behavior is the likelihood is monotonous (From wiki: "The Karlin-Rubin theorem extends the Neyman–Pearson lemma to settings involving composite hypotheses with monotone likelihood ratios.")

Later, helped with example 2, it becomes clear that the optimality is conditional on the model being correctly specified. There could be a setting where majority voting from the multiple predictors could be the best predictor. If there are interactions, SVM could work better than linear regression...

estimable contrast

 An old technical document from sas captures the relationship among Type I, II and III sum of squares.

qq plot interpretation

SAS doc gives the references and a few simple rules, and R community gives more examples around these rules. I think a better way to remember these associations is to understand the mechanism. In this way, wikipedia is doing a better job:

A simple case is where one has two data sets of the same size. In that case, to make the Q–Q plot, one orders each set in increasing order, then pairs off and plots the corresponding values

So it is almost a scatter plot of observed value vs simulated value from a distribution. Still it is not always easy to interpret, especially from a small dataset

Besides, the qqPlot function from r package 'car' includes CIs for observed quantiles by default, or more specialized qqplot for log10(p-values) from pQQ function out of Haplin

consistency property of lasso

This is about the meaning of | as in the "irrepresentable condition"  from On Model Selection Consistency of Lasso

It seems the operation is |x| = abs(sum(x)). I have never seen such a norm. But their Section "3.1 Simulation Example 1" makes me think so: 

... in two settings: (a) β1 = 2, β2 = 3 ; and (b) β1 =  2, β2 = 3. In both settings, X(1) = (X1;X2), X(2) = X3 and through (2), it is easy to get C21inv(C11)= (2/3,2/3). Therefore Strong Irrepresentable Condition fails for setting (a) and holds for setting (b).

Their Proof for Corollary 1 makes me think so:

There is a similar use here.

oracle

In "Sparse Estimators and the Oracle Property, or the Return of Hodges’ Estimator": 

Fan and Li (2001), the SCAD estimator, with appropriate choice of the regularization (tuning) parameter, possesses a sparsity property, i.e., it estimates zero components of the true parameter vector exactly as zero with probability approaching one as sample size increases while still being consistent for the non-zero components...In other words, with appropriate choice of the regularization parameter, the asymptotic distribution of the SCAD estimator based on the overall model and that of the SCAD estimator derived from the most parsimonious correct model coincide. Fan and Li (2001) have dubbed this property the “oracle property”....It is well-known for Hodges’ estimator that the maximal (scaled) mean squared error grows without bound as sample size increases (e.g., Lehmann and Casella (1998), p.442), whereas the standard maximum likelihood estimator has constant finite quadratic risk. In this note we show that a similar unbounded risk result is in fact true for any estimator possessing the sparsity property. This means that there is a substantial price to be paid for sparsity even though the oracle property (misleadingly) seems to suggest otherwise. 

In "Modern statistical estimation via oracle inequalities":

Theorem 4.1. The James–Stein estimate obeys .

In other words, the James–Stein estimator is almost as good as the ideal estimator in a mean-squared error sense.The inequality (4.2) is an oracle inequality. An oracle inequality relates the performance of a real estimator with that of an ideal estimator which relies on perfect information supplied by an oracle, and which is not available in practice.

mean, median and mode

this page explains a mean minimizes the  norm of the residual: 
a median minimizes its  norm and a mode minimizes the zero norm of the residual, namely .See the wikipedia page about median.

from here, it was further explained that

Inder Jeet Taneja’s book draft has a nice survey of the results: if you fix the upper and lower boundary, and maximize entropy, you’ll get the uniform distribution. If you fix the mean and the expected L2 norm (d^2) between the mean and the distribution, maximizing the entropy you’ll get the Gaussian. If you fix the expected L1 norm (|d|) between the mean and the distribution, maximizing the entropy you’ll get the Laplace (also referred to as Double Exponential). Moreover, log(1+d^2) norm will yield the Cauchy distribution – a special case of the standard heavy-tailed Student distribution.

interaction coding

In sas: we have

Nested Effects

Nested effects are generated in the same manner as crossed effects. Hence, the design columns generated by the following statements are the same (but the ordering of the columns is different):

model y=a b(a);		(B nested within A)
model y=a a*b;		(omitted main effect for B)

seriation

Seriation, also the name for an R package 

is to arrange all objects in a set in a linear order given available data and some loss or merit function in order to reveal structural information. Together with cluster analysis and variable selection, seriation is an important problem in the field of combinatorial data analysis

This post discussed the application of seriation on plotting binary matrices, together with other clustering algorithms.

Prioritizing GWAS Results

This review paper mentioned 3 analytic methods to prioritizing GWAS results (for follow-up studies):

• Meta-analysis seeks to pool information from multiple GWAS to increase the chances of finding true positives among the false positives and provides a way to combine associations across GWAS, even when the original data are unavailable.

• Testing for epistasis within a single GWAS study can identify the stronger results that are revealed when genes interact.

• Pathway analysis of GWAS results is used to prioritize genes and pathways within a biological context. Following a GWAS, association results can be assigned to pathways and tested in aggregate with computational tools and pathway databases.

The default plot function from glmnet is quite basic and plain. The following function tries to improve the plots

#this function is to achieve the same purpose of plot.glmnet()
#with the following enhancement:

# 1) show variable number instead of column number
# 2) show the value of lambda that gives smallest cross validation error with a solid line
# 3) show the largest lambda such that the error is within 1 se of the minimal cross validation error with a dashed line
 
#input variables: a glmnet object from glmnet(), a cv. glmnet object from cv.glmnet(), and a possible plot title
 

igraph tutorial

http://horicky.blogspot.com/2012/04/basic-graph-analytics-using-igraph.html
http://igraph.sourceforge.net/igraphbook/
leaves = V(g)[degree(g, mode="out")==0]

from here:

The methods under consideration included observed case MMRM, per protocol visits MMRM, interval last observation carried forward (LOCF) MMRM, and a hybrid of the per protocol visits and interval LOCF MMRM approaches.Simulation results reveal that the method that best controls the type I error rate is the per protocol visits method.

mean corr is equal to sample corr

at least for independent pairs. For prove see here.

Hello, Bugs world!

Here is my first Bayesian code modified from the sample code for the Chapter 3 of Bayesian Adaptive Methods for Clinical Trials. The R code depends on BRugs that interfaces with Openbugs (manual is here).

Comparison of general and generalized linear models

wiki

conditional exact logistic regression

Conditional Logistic Regression:

Taking the stratification into account by "conditioning out" (and not estimating) the stratum-specific intercepts gives consistent and asymptotically normal MLEs for the slope coefficients. If your nuisance parameters are not just stratum-specific intercepts, you can perform an exact conditional logistic regression.

The likelihood function to be maximized feels really like that in proc phreg with a discrete option. Each item (for a stratum) in the likelihood function is the conditional probability of observing subject 1 to h who actually had an event conditioning on h event happened (for any h subjects) in the stratum. The concept of sufficient statistic is not explicitly. Conditional logistic regression is called with the strata statement. More examples are from here and here.

Exact Conditional Logistic Regression
Exact conditional inference is based on generating the conditional distribution for the sufficient statistics of the parameters of interest. This distribution is called the permutation or exact conditional distribution. One example from sas suggests that we move from conditional regression to exact conditional regression when "you believe the data set is too small or too sparse for the usual asymptotics to hold.":

• by "sparse", they probably refer to the problem of complete separation of quasi-complete separation, and the exact method is one way to handle this, especially when the firth option is only available in sas 9.1.3;
• by "usual asymptotics", they probably refer to the issue of  large-sample asymptotic normality when we do not have a large sample size compared to the number of parameters to estimate.

Geometric Statistics, geometric CV, intra-subject variation

found from onbiostatistics and here. Additional discussion can be found here.
also for a log normal variable x, there are

y=log(x)~N(mu, sigma**2)
E(x)=exp(mu+sigma**2/2) and
sd(x)=exp(mu+sigma**2/2)*sqrt(exp(sigma**2)-1)

cv(x)=sqrt(exp(sigma**2)-1)

cross product

in R, cross product is defined similar to inner product, but is different in other areas.

built-in multiple testing mechanism in SAS

proc multtest by Peter H. Westfall and Russell D. Wolfinger
the adjust option in proc mixed, or proc glm
exact statement from PROC NPAR1WAY